Torsion classes generated by silting modules

TL;DR

This paper characterizes torsion classes generated by silting modules over certain rings, linking them to special preenvelopes and minimal silting modules, and explores dual results involving cosilting modules.

Contribution

It establishes a precise correspondence between torsion classes generated by silting modules and special preenvelopes over hereditary or perfect rings, and extends dual results to cosilting modules.

Findings

Torsion classes generated by silting modules are characterized by special preenvelopes.

Every torsion enveloping class in Mod-R is generated by a minimal silting module.

Dual results identify covering torsion-free classes as those cogenerated by cosilting modules.

Abstract

We study the classes of modules which are generated by a silting module. In the case of either hereditary or perfect rings it is proved that these are exactly the torsion $\mathcal{T}$ such that the regular module has a special $\mathcal{T}$-preenvelope. In particular every torsion enveloping class in $\textrm{Mod-} R$ are of the form $\mathrm{Gen}(T)$ for a minimal silting module $T$. For the dual case we obtain for general rings that the covering torsion-free classes of modules are exactly the classes of the form $\mathrm{Cogen}(T)$, where $T$ is a cosilting module.